Steel Frame Design Manual

Steel Frame Design Manual Eurocode 3-1:2005 with 8:2004

Eurocode 3-1:2005 with Eurocode 8:2004 Steel Frame Design Manual for ETABS 2016 ISO ETA122815M13 Rev 0 Proudly developed in the United States of America July 2016

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DISCLAIMER CONSIDERABLE TIME, EFFORT AND EXPENSE HAVE GONE INTO THE DEVELOPMENT AND TESTING OF THIS SOFTWARE. HOWEVER, THE USER ACCEPTS AND UNDERSTANDS THAT NO WARRANTY IS EXPRESSED OR IMPLIED BY THE DEVELOPERS OR THE DISTRIBUTORS ON THE ACCURACY OR THE RELIABILITY OF THIS PRODUCT. THIS PRODUCT IS A PRACTICAL AND POWERFUL TOOL FOR STRUCTURAL DESIGN. HOWEVER, THE USER MUST EXPLICITLY UNDERSTAND THE BASIC ASSUMPTIONS OF THE SOFTWARE MODELING, ANALYSIS, AND DESIGN ALGORITHMS AND COMPENSATE FOR THE ASPECTS THAT ARE NOT ADDRESSED. THE INFORMATION PRODUCED BY THE SOFTWARE MUST BE CHECKED BY A QUALIFIED AND EXPERIENCED ENGINEER. THE ENGINEER MUST INDEPENDENTLY VERIFY THE RESULTS AND TAKE PROFESSIONAL RESPONSIBILITY FOR THE INFORMATION THAT IS USED.

Contents Chapter 1 Introduction 1.1 Units 1-2 1.2 Axes Notation 1-2 1.3 Symbols 1-2 Chapter 2 Assumptions and Limitations 2.1 Assumptions 2-1 2.1.1 General 2-1 2.1.2 Axial Force Check 2-2 2.1.3 Bending Moment Check 2-2 2.1.4 Shear Force Check 2-2 2.1.5 Combined Force Check 2-2 2.2 Limitations 2-4 2.2.1 General 2-4 2.2.2 Axial Force Check 2-4 2.2.3 Torsional Moment Check 2-5 2.2.4 Combined Force Check 2-5 Contents i

Steel Frame Design Eurocode 3-2005 Chapter 3 Chapter 4 Design Flow Charts General Design Parameters 4.1 Partial Factors 4-1 4.2 Design Forces 4-1 4.3 Design Load Combinations 4-2 4.3.1 Ultimate Strength Combinations 4-2 4.3.2 Serviceability Combinations 4-4 4.4 Material Properties 4-4 4.5 Section Classification 4-5 Chapter 5 Design for Axial Forces 5.1 Axial Area 5-1 5.2 Tension Check 5-2 5.3 Compression Check 5-2 5.4 Axial Buckling Check 5-3 5.5 Member Unsupported Lengths 5-5 5.6 Effective Length Factor (K) 5-8 Chapter 6 Design for Bending Moment 6.1 Moment Check 6-1 Contents ii

Contents 6.2 Lateral-Torsional Buckling Check 6-4 Chapter 7 Design for Shear Force 7.1 Shear Area 7-1 7.2 Shear Check 7-1 7.3 Shear Buckling Check 7-2 Chapter 8 Design for Torsion 8.1 Analysis of Torsion Stresses 8-1 8.2 Shear Check in Presence of Torsion 8-2 8.3 Design for Combined Actions 8-3 Chapter 9 Design for Combined Forces 9.1 Design for Cross-Section Resistance 9-2 9.1.1 Bending, Axial Force, and Shear Check 9-2 9.1.2 Members Subjected to Shear Force 9-7 9.2 Design for Buckling Resistance of Members 9-8 9.2.1 Class 1, 2, and 3 Sections Under Flexure and Axial Compression 9-8 9.2.2 Class 4 Sections Under Flexure and Axial Compression 9-9 9.2.3 Class 1, 2, and 3 Sections Under Flexure and Axial Tension 9-10 9.2.4 Class 4 Sections Under Flexure and Axial Tension 9-10 Chapter 10 Special Seismic Provisions 10.1 Design Preferences 10-1 Contents iii

Steel Frame Design Eurocode 3-2005 10.2 Overwrites 10-2 10.3 Supported Framing Types 10-2 10.4 Member Design 10-3 10.4.1 Ductility Class High Moment-Resisting Frames (DCH MRF) 10-4 10.4.2 Ductility Class Medium Moment-Resisting Frames (DCM MRF) 10-6 10.4.3 Ductility Class Low Moment-Resisting Frames (DCL MRF) 10-6 10.4.4 Ductility Class High Concentrically Braced Frames (DCH CBF) 10-7 10.4.5 Ductility Class Medium Concentrically Braced Frames (DCM CBF) 10-9 10.4.6 Ductility Class Low Concentrically Braced Frames (DCL CBF) 10-9 10.4.7 Ductility Class High Eccentrically Braced Frames (DCH EBF) 10-10 10.4.8 Ductility Class Medium Eccentrically Braced Frames (DCM EBF) 10-13 10.4.9 Ductility Class Low Eccentrically Braced Frames (DCL EBF) 10-14 10.4.10 Inverted Pendulum 10-14 10.4.11 Secondary 10-15 10.5 Design of Joints Components 10-15 Appendix A Design Preferences Appendix B Design Overwrites 10.5.1 Design of Continuity Plates 10-16 10.5.2 Design of Supplementary Web Plates 10-22 10.5.3 Weak Beam/Strong Column Measure 10-26 10.5.4 Evaluation of Beam Connection Shears 10-28 10.5.5 Evaluation of Brace Connection Forces 10-29 Contents iv

Contents Appendix C Nationally Determined Parameters (NDPs) References Contents v

Chapter 1 Introduction This manual describes the steel frame design algorithms in the software for the Eurocode 3-2005 [EN 1993-1-1] design code. The design algorithms in the software for Eurocode 3 cover strength checks, as detailed in this manual. Requirements of the code not documented in this manual should be considered using other methods. The default implementation in the software is the CEN version of the code. Additional country specific National Annexes are also included. The Nationally Determined Parameters are noted in this manual with [NDP]. Changing the country in the Design Preferences will set the Nationally Determined Parameters for the selected country as defined in Appendix C. It is important to read this entire manual before using the design algorithms to become familiar with any limitations of the algorithms or assumptions that have been made. For referring to pertinent sections of the corresponding code, a unique prefix is assigned for each code. Reference to the EN 1993-1-1:2005 code is identified with the prefix EC3. Reference to the EN 1993-1-5:2006 code is identified with the prefix EN 1993-1-5. 1-1

Steel Frame Design Eurocode 3-2005 Reference to the ENV 1993-1-1:1992 code is identified with the prefix EC3-1992. Reference to the Eurocode 1990:2002 code is identified with the prefix EC0. 1.1 Units The Eurocode 3 design code is based on Newton, millimeter, and second units and, as such, so is this manual, unless noted otherwise. Any units, imperial, metric, or MKS may be used in the software in conjunction with Eurocode 3 design. 1.2 Axes Notation The software analysis results refer to the member local axes system, which consists of the 2-2 axis that runs parallel to the web and the 3-3 axis that runs parallel to the flanges. Therefore, bending about the 2-2 axis would generate minor axis moment, and bending about the 3-3 axis would generate major axis moment. The Eurocode 3 design code refers to y-y and z-z axes, which are equivalent to the software 3-3 and 2-2 axes, respectively. These notations may be used interchangeably in the design algorithms, although every effort has been made to use the design code convention where possible. 1.3 Symbols The following table provides a list of the symbols used in this manual, along with a short description. Where possible, the same symbol from the design code is used in this manual. a Torsional bending constant. It is given by a = ( EIw) ( GIT ), where EIw represents the warping stiffness and GIT is the St Venant torsional stiffness. It generally expresses the rate at which the warping torsional moment diminishes from a position where warping is restrained, mm 1-2 Units

Chapter 1 Introduction A Gross area of cross section, mm 2 A net Net area of cross section, mm 2 A v Shear area, mm 2 A w Web area, mm 2 b C 1 Width of the section, mm Moment diagram factor E Modulus of elasticity, N/mm 2 f u Steel ultimate strength, N/mm 2 f y Steel yield strength, N/mm 2 f yw Steel yield strength of the web, N/mm 2 F w, Design shear force in the flange due to warping moment, N G Shear modulus. It is given by G E 21 ( ) h h w = +υ, where E is the modulus of elasticity and υ is the Poisson s ratio. For structural steel EG = 26. and G 81, 000, N/mm 2 Depth of the section, mm Web height, mm I Moment of inertia, mm 4 I F I T I w k yy, k zz, k yz, k zy L cr M b,rd M c,rd Moment of inertia of a flange of an I beam about its own center of gravity or about the minor axis of bending of the whole section, I = t b 3 12, mm 4 F f f St. Venant torsional constant. It is the section property relating St. Venant torsional moment to the first derivative of rotation (twist per unit length). In many texts it is referred to as J, mm 4 Warping constant. It is the section property relating the warping torsional moment to the third derivative of rotation. In many texts it is referred to as H or C w, mm 6 Interaction factors Buckling length, mm Design buckling resistance moment, N-mm Design bending resistance, N-mm Symbols 1-3

Steel Frame Design Eurocode 3-2005 M M el,rd M pl,rd M Rk M w, M y,v,rd M z,t, N b,rd N cr N c,rd N N pl,rd N Rk N t,rd N u,rd t f t w T T t, T w. V c,rd V b,rd V bf,rd V bw,rd V Design bending moment, N-mm Elastic design bending resistance, N-mm Plastic design bending resistance, N-mm Characteristic bending resistance, N-mm Warping moment. It is the bending moment in a flange acting as a result of restraint of warping. The moment at the two flanges are equal and of opposite sign, N-mm Reduced design bending resistance accounting for shear, N-mm Additional bending moment about the minor axis caused by the rotation of the beam and torsion, N-mm Design buckling resistance, N Elastic critical force, N Design compression resistance, N Design axial force, N Plastic design axial resistance, N Characteristic compression resistance, N Design tension resistance, N Design ultimate tension resistance, N Flange thickness, mm Web thickness, mm Design value of total torsional moment, N-mm Design value of internal St Venant torsional moment, N-mm Design value of internal warping torsional moment, N-mm Design shear resistance, N Design shear buckling resistance, N Flange contribution of the design shear buckling resistance, N Web contribution of the design shear buckling resistance, N Design shear force, N 1-4 Symbols

Chapter 1 Introduction V pl,rd V pl,t,rd Plastic design shear resistance, N Reduced design plastic shear resistance making allowance for the presence of a torsional moment, N W el,min Minimum elastic section modulus, mm 3 W pl Plastic section modulus, mm 3 α, α LT Imperfection factor χ χ LT χ w Reduction factor for buckling Reduction factor for lateral-torsional buckling Web shear buckling contribution factor ε Coefficient dependent on f y, ε= 235 f y φ φ ' φ '' The angle of rotation of a cross-section about the longitudinal axis, rad The first derivative of the angle of rotation of a cross-section about the longitudinal axis, ' d dx φ = φ, rad/mm The second derivative of the angle of rotation of a crosssection about the longitudinal axis, φ '' = d φ dx, rad/mm 2 2 2 φ ''' The third derivative of the angle of rotation of a crosssection about the longitudinal axis, φ ''' = d φ dx, rad/mm 3 3 3 Φ Φ LT γ M0 γ M1 γ M2 η λ λ LT Value for calculating the reduction factor χ Value for calculating the reduction factor χ LT Partial factor for resistance of cross-sections Partial factor for resistance of members to instability Partial factor for resistance of cross-sections in tension to fracture Factor for shear area Non-dimensional slenderness Non-dimensional slenderness for lateral-torsional buckling λ LT,0 Plateau length of the lateral-torsional buckling curves Symbols 1-5

Steel Frame Design Eurocode 3-2005 λ w ρ ρ T Slenderness parameter Reduction factor accounting for shear forces Reduction factor for design plastic shear resistance making allowance for the presence of a torsional moment. It is takes ρ = V V. as T pl,t,rd pl,rd τ Design shear stress at a point in a section, N/mm 2 τ Design shear stress due to St Venant torsion, N/mm 2 t, τ Design shear stress due to warping torsion, N/mm 2 w, ψ Ratio of moments in a segment 1-6 Symbols

Chapter 2 Assumptions and Limitations This chapter describes the assumptions made and the limitations of the design algorithm for the Eurocode 3-2005 steel frame design. All of the assumptions and limitations should be reviewed before using the design algorithm. 2.1 Assumptions The assumptions made in the design algorithm are listed in the following sections, along with a description of how they may affect the design results. 2.1.1 General The following assumptions apply generically to the design algorithm. The analysis model geometry, properties, and loads adequately represent the building structure for the limit states under consideration (EC3 5.1.1). It is assumed that the steel grades used adhere to Eurocode 3-2005, Table 3.1 or an associated National Annex (EC3 3.1(2)). The acceptable use of other materials shall be independently verified. The automated load combinations are based on the STR ultimate limit states and the characteristic serviceability limit states. 2-1

Steel Frame Design Eurocode 3-2005 2.1.2 Axial Force Check The following assumptions apply to the axial force check. Hot rolled tubular sections are assumed to be hot finished for selecting the appropriate buckling curve from EC3 Table 6.2. This is nonconservative if cold formed sections are used. For welded Box sections, if bt f < 30 and ht w < 30, it is assumed that the weld thickness is more than 0.5t f (EC3 Table 6.2). 2.1.3 Bending Moment Check The following assumptions apply to the bending moment check. The load is assumed to be applied at the shear center for the calculation of the elastic critical moment. Any eccentric moment due to load applied at other locations is not automatically accounted for. 2.1.4 Shear Force Check The following assumptions apply to the shear force check. Plastic design is assumed such that V c,rd is calculated in accordance with EC3 6.2.6(2). Transverse stiffeners exist only at the supports and create a non-rigid end post for the shear buckling check. No intermediate stiffeners are considered. The contribution from the flanges is conservatively ignored for the shear buckling capacity. 2.1.5 Torsional Moment Check The following assumptions apply to the shear force check. Total torsional moment is split into a St Venant torsion T t, and a warping torsion T w,. 2-2 Assumptions

Chapter 2 Assumptions and Limitations The warping torsion, M w,, contributes to the bending moment in the flanges. This affects the interaction checks. It also increases the minor axis bending moment by a small amount, M. The interaction equations are z,t, modified to accommodate the warping moment, minor axis bending moment, M z,t, M w, and additional. The following equations are modified: Eq. 6.2 of section EC3 6.2.1(7), Eq. 6.41 of section EC3 6.2.9.1(6), Eq. 6.44 of section EC3 6.2.9.3(2). One additional interaction equation is checked which is given in Annex A of BS EN 1993-6. The effect of combined shear force and torsional moment in the reduction of section shear resistance is evaluated by using the section EC3 6.2.7(9). The elastic verification using the yield criterion given in EC3 6.2.1(5) is not used as this criterion is conservative. This criterion ignores the plastic stress redistribution. In doing torsional moment check the book Design of Steel Beams in Torsion in Accordance with Eurocodes and UK National Annexes, the SCI Publication P385, is taken as the guide (Hughes, Iles, and Malik, 2011). 2.1.6 Combined Forces Check The following assumptions apply to the combined forces check. The interaction of bending and axial force is checked for certain sections (shapes) and for certain classes of sections, in accordance with EC3 6.2.1(7), which may be conservative compared to EC3 6.2.9, when no special clause in EC3 6.2.9 is applicable to that shape and that class. The warping torsion, M w,, contributes to the bending moment in the flanges. This affects the interaction checks. It also increases the minor axis bending moment by a small amount, M z,t,. The interaction equations are modified to accommodate the warping moment, M w, and additional minor axis bending moment, M z,t,. The following equations are modified: Eq. 6.2 of section EC3 6.2.1(7), Eq. 6.41 of section EC3 6.2.9.1(6), Eq. 6.44 of section EC3 6.2.9.3(2). One additional interaction equation is checked which is given in Annex A of BS EN 1993-6. Assumptions 2-3

Steel Frame Design Eurocode 3-2005 2.2 Limitations The limitations of the design algorithm are listed in the following sections, along with a work around where possible. 2.2.1 General The following limitations apply generically to the design algorithm. Sections with a material thickness, t < 3 mm are not designed (EC3 1.1.2(1)). The special requirements in accordance with EN 1993-1-3 for cold formed thin gauge members are not covered in this implementation (EC3 1.1.2(1)). The material yield is not adjusted based on the thickness of the section. Different material properties should be defined for sections of different thickness if the thickness affects the material yield value (EC3 3.2.1, Table 3.1). The effects of torsion are not considered in the design (EC3 6.2.7) and should be considered using other methods. The special requirements in accordance with Eurocode EN 1993-1-12 for high-strength steels above S460 currently are not considered. The special requirements in accordance with EN 1993-1-6 for Circular Hollow (tube) sections with Class 4 cross-sections are not covered in this implementation (EC3 6.2.2.5(5)). 2.2.2 Axial Force Check The following limitations apply to the axial force check. The net area is not determined automatically. This can be specified on a member-by-member basis using the Net Area to Total Area Ratio overwrite. 2-4 Limitations

2.2.3 Torsional Moment Check Chapter 2 Assumptions and Limitations The following limitations apply to the torsional moment check. The torsion check is limited to the following open section: Doublysymmetric I-shapes. I-shapes with single symmetric sections are not checked for torsion. The torsion check is limited to the following closed sections: box and pipe shapes. 2.2.4 Combined Forces Check The following limitations apply to the combined forces checks. The code allows the engineers to design the cross-sections with Class 3 web and Class 1 or 2 flanges as a Class 2 cross-section with an effective web area as specified in EC3 6.2.2.4 (EC3 5.5.2(11), EC3 6.2.2.4(1)). However, the program does not take this advantage, which is conservative. Limitations 2-5

Chapter 3 Design Flow Charts The flow charts on the following pages provide a pictorial representation of the design algorithm for Eurocode 3-2005 steel frame design. These flow charts provide a summary of the steps taken and the associated code clauses used. Additional detailed information defining the steps used in the algorithm is provided in the chapters that follow. The following flow charts are provided: member design design axial resistance design axial buckling resistance design bending resistance design lateral-torsional buckling resistance design shear resistance The flowcharts given this this chapter, in general, assume that there is no consideration for torsion. However, the program, can optionally do design checks for certain sections (I, Box, and Pipes) considering torsion. In doing so, the program uses modified versions of the interaction equations. This has been discussed in the appropriate chapters in details. 3-1

Steel Frame Design Eurocode 3-2005 Start EC3 Table 5.2 Determine section class Class 1, 1, 2, 2, 3, or or 3 4 Yes No Too Class Slender 4 not designed Check shear capacity V V Rd See Figure 3.6 End Check bending capacity M min(m Rd, M b,rd ) See Figures 3.4 and 3.5 Check axial capacity N min(n Rd, N b,rd ) See Figures 3.2 and 3.3 Check force interaction criteria EC3 6.2.1(7), EC3 6.2.1(7), 6.2.9, 6.3.3 6.3.3 Critical Critical utilization utilization ratio, ratio, error error and and warning messages End 3-2 Member Design

Chapter 3 Design Flow Charts Figure 3-1 Member Design Start Tension or compression EC3 6.2.3(2) Tension Compression EC3 6.2.4(2) Calculate design tension resistance N t,rd = min(n pl,rd, N u,rd ) Calculate design compression resistance N c,rd Design Axial Resistance N Rd End Figure 3-2 Design Axial Resistance Axial Resistance 3-3

Steel Frame Design Eurocode 3-2005 Start Calculate elastic critical force N cr Critical force N cr EC3 6.3.1.2(1) Calculate non-dimensional slenderness λ Non-dimensional slenderness λ EC3 Table 6.1, Table 6.2 No λ 0.2 χ = 1.0 Yes EC3 6.3.1.2(1) Determine buckling curve Buckling curve and factors EC3 6.3.1.1(3) Calculate reduction factor χ Reduction factor χ Calculate design axial buckling resistance N b,rd Design buckling resistance N b,rd End Figure 3-3: Design Axial Buckling Resistance 3-4 Axial Buckling Resistance

Chapter 3 Design Flow Charts Start EC3 Table 5.2 Determine section class Class 1 or 2 Yes No Calculate design moment resistance M = M c,rd pl,rd Class 3,4 3 Yes No EC3 6.2.5(2) Calculate design moment resistance M c,rd = M or el,rd M = w f γ c,rd eff min y mo Too Class Slender 4 not designed Design Moment Resistance M c,rd End EC3 6.2.5(2) 6.2.8(3), 6.2.8(5) Calculate Design Design Moment Moment Resistance Resistance M c,rd v,rd EC3 6.2.5(2) 6.2.9.1 Calculate Reduced Design Design Moment Moment Resistance Resistance M c,rd N,Rd End Figure 3-4: Design Moment Resistance Moment Resistance 3-5

Steel Frame Design Eurocode 3-1:2005 EC3 6.3.2.2(2) EC3 EC3-1193 6.2.6(2) F1.1 Start Calculate critical buckling moment M cr Critical buckling moment M cr EC3 6.3.2.2(1) 6.2.6(2) Calculate non-dimensional slenderness λ LT Non-dimensional slenderness λ LT EC3 Table 6.2.6(2) 6.3 Table 6.4 No λ λ LT LT,0 Yes Ignore LTB EC3 6.3.2.2(1) 6.2.6(2) Determine buckling curve Buckling curve and factors End EC3 6.3.2.1(3) 6.2.6(2) Calculate reduction factor χ LT Reduction factor χ LT Calculate design buckling resistance moment M b,rd Design Buckling Resistance M b,rd End Figure 3-5: Design Buckling Resistance 3-6 Buckling Resistance

Chapter 3 Design Flow Charts Start EC3 6.2.6(6) h w ε 72 t η w EC3 6.2.6(2) No Yes EC3-1-5 5.2(1), 5.3(1) Calculate design shear resistance Calculate shear buckling resistance V c,rd = V pl,rd V c,rd = V b,rd Design Shear Resistance V c,rd End Figure 3-6: Design Shear Resistance Shear Resistance 3-7

Chapter 4 General Design Parameters This chapter provides a detailed description of the implementation of the various parameters used in the design algorithm for the Eurocode 3-2005 steel frame design. These parameters are subsequently used in the following chapters for the design of sections for the applied force actions. 4.1 Partial Factors The following partial factors, γ M, are applied to the various characteristic resistance values determined in the following chapters. The partial factor values may be overwritten in the Design Preferences. γ M 0 = 1.00 [NDP] (EC3 6.1(1)) γ M1 = 1.00 [NDP] (EC3 6.1(1)) γ M 2 = 1.25 [NDP] (EC3 6.1(1)) 4.2 Design Forces The following design force actions are considered in the design algorithm covered in the following chapters. The force actions are determined using the 4-1

Steel Frame Design Eurocode 3-2005 appropriate load combinations described in the following section. Axial force (tension or compression), N Shear force (major or minor axis), V Bending moment (major or minor axis), M Torsion, T It should be noticed that the torsion check is only limited to doubly-symmetric I-shapes, box shapes, and pipe shapes. 4.3 Design Load Combinations The design load combinations are combinations of load cases for which the structure is designed and checked. A default set of automated load combinations is available in the software, as described in this section. These default combinations can be modified or deleted. In addition, manually defined combinations can be added should the default combinations not cover all conditions required for the structure of interest. The default load combinations considered by the software for Eurocode 3-2005, are defined in the following sections and handle dead (D), live (L), wind (W), and earthquake (E) loads. For other load types, combinations should be manually generated. The following two sections describe the automated load combinations generated by the software for ultimate strength and serviceability, in accordance with Eurocode 1990:2002 [EN 1990:2002]. 4.3.1 Ultimate Strength Combinations Eurocode 0:2002 allows load combinations to be defined based on EC0 equation 6.10 or the less favorable EC0 equations 6.10a and 6.10b [NDP]. γ G +γ P+γ Q + γ ψ Q (EC0 Eq. 6.10) G, j k, j p Q,1 k,1 Qi, 0, i ki, j 1 i> 1 4-2 Design Load Combinations

Chapter 4 General Design Parameters γ G +γ P+γ ψ Q + γ ψ Q (EC0 Eq. 6.10a) G, j k, j p Q,1 Q,1 k,1 Qi, 0, i ki, j 1 i> 1 ξ γ G +γ P+γ Q + γ ψ Q (EC0 Eq. 6.10b) j G, j k, j p Q,1 k,1 Qi, 0, i ki, j 1 i> 1 Load combinations including earthquake effects are generated based on: G + P+ A + ψ Q (EC0 Eq. 6.12b) k, j 2, i k, i j 1 i> 1 The following load combinations are considered if the option is set to generate the combinations based on EC0 equation 6.10. γ Gj,sup D (EC0 Eq. 6.10) γ Gj,sup D + γ Q,1 L (EC0 Eq. 6.10) γ Gj,inf D ± γ Q,1 W γ Gj,sup D ± γ Q,1 W (EC0 Eq. 6.10) γ Gj,sup D + γ Q,1 L ± γ Q,i ψ 0,i W (EC0 Eq. 6.10) γ Gj,sup D ± γ Q,1 W + γ Q,i ψ 0,i L D ± 1.0E (EC0 Eq. 6.12b) D ± 1.0E + ψ 2,i L The following load combinations are considered if the option is set to generate the combinations based on the maximum of EC0 equations 6.10a and 6.10b. γ Gj,sup D ξγ Gj,sup D γ Gj,sup D + γ Q,1 ψ 0,1 L ξγ Gj,sup D + γ Q,1 L γ Gj,inf D ± γ Q,1 ψ 0,1 W γ Gj,sup D ± γ Q,1 ψ 0,1 W γ Gj,inf D ± γ Q,1 W ξγ Gj,sup D ± γ Q,1 W γ Gj,sup D + γ Q,1 ψ 0,1 L ± γ Q,i ψ 0,i W (EC0 Eq. 6.10a) (EC0 Eq. 6.10b) (EC0 Eq. 6.10a) (EC0 Eq. 6.10b) (EC0 Eq. 6.10a) (EC0 Eq. 6.10b) (EC0 Eq. 6.10a) Design Load Combinations 4-3

Steel Frame Design Eurocode 3-2005 γ Gj,sup D ± γ Q,1 ψ 0,1 W + γ Q,i ψ 0,i L ξγ Gj,sup D + γ Q,1 L ± γ Q,i ψ 0,i W ξγ Gj,sup D ± γ Q,1 W + γ Q,i ψ 0,i L D ± 1.0E D ± 1.0E + ψ 2 i L (EC0 Eq. 6.10b) (EC0 Eq. 6.12b) The variable values and factors used in the load combinations are defined as: γ Gj,sup = 1.35 [NDP] γ Gj,inf = 1.00 [NDP] (EC0 Table A1.2(B)) (EC0 Table A1.2(B)) γ Q,1 = 1.5 [NDP] (EC0 Table A1.2(B)) 0.7 (live load, nots torage) ψ 0, i = [NDP] (EC0 Table A1.1) 0.6 (wind load) ξ = 0.85 [NDP] (EC0 Table A1.2(B)) ψ 2,i = 0.3 (assumed office/residential) [NDP] (EC0 Table A1.1) 4.3.2 Serviceability Combinations The following characteristic load combinations are considered for the deflection checks. D D + L (EC0 Eq. 6.10a) (EC0 Eq. 6.10a) 4.4 Material Properties The nominal values of the yield strength f y and ultimate strength f u are used in the design. The design assumes that the input material properties conform to the steel grades listed in the code (EC3 Table 3.1) or have been verified using other methods, to be adequate for use with Eurocode 3-2005. The design values of material coefficients (EC3 3.2.6) are taken from the input material properties, rather than directly from the code. 4-4 Material Properties

4.5 Section Classification Chapter 4 General Design Parameters Eurocode 3-2005 classifies sections into four different classes, which identify the extent to which the resistance and rotation capacity is limited by local buckling. The different classes are based on the width-to-thickness ratio of the parts subject to compression and are defined as: Class 1 section can form a plastic hinge with the rotation capacity required from plastic analysis, without reduction of the resistance. Class 2 section can develop its plastic moment capacity, but has limited rotation capacity. Class 3 section in which the stress in the extreme compression fiber of the section, assuming an elastic distribution of stresses, can reach the yield strength, but local buckling is likely to prevent the development of the plastic moment capacity. Class 4 section is subject to local buckling before reaching the yield stress in one or more of the parts. Too Slender section does not satisfy any of the criteria for Class 1, 2, 3, or 4. This happens when t f < 3 mm or t w < 3 mm. Too Slender sections are beyond the scope of the code. They are not checked/designed. The following three tables identify the limiting width-to-thickness ratios for classifying the various parts of the cross-section, subject to bending only, compression only, or combined bending and compression. The various parameters used in calculating the width-to-thickness ratio limits are defined as: ε= 235 f y (EC3 Table 5.2) N ψ= 1+ 2, 3.0 <ψ 1.0 Af y (EC3 5.5.2, Table 5.2) for I-sections, Channels: 1 h 1 N α= ( tf r), 1 1 c α 2 2 twf y (EC3 5.5.2, Table 5.2) Section Classification 4-5

Steel Frame Design Eurocode 3-2005 for Boxes and Double Channel sections 1 h 1 N α= ( tf r), 1 1 c + α 2 4 twf y (EC3 5.5.2, Table 5.2) Table 4-1: Width-To-Thickness Ratios - Bending Only Shape Part Ratio Class 1 Class 2 Class 3 I-sections, Channels Web c/t 72ε 83ε 124ε Tees Web c/t If tip is in comp. 9ε /α If tip is in tension, α 9ε α If tip is in comp. 10ε /α If tip is in tension, α 10ε α 21ε k σ Flange c/t 9ε 10ε 14ε Boxes Web, flange c/t 72ε 83ε 124ε Tubes/Pipes Wall d/t 50ε 2 70ε 2 90ε 2 Solid Bars Bar N/A Assumed to be Class 2 General, Section Designer Section N/A Assumed to be Class 3 4-6 Section Classification

Chapter 4 General Design Parameters Table 4-2: Width-To-Thickness Ratios - Compression Only Shape Part Ratio Class 1 Class 2 Class 3 I-sections, Channels Web c/t 33ε 38ε 42ε Flange c/t 9ε 10ε 14ε Tees Web, flange c/t 9ε 10ε 14ε Angles, Double Angles Legs h/t and (b+h)/2t N/A N/A 15ε and 11.5ε Boxes Web, flange c/t 33ε 38ε 42ε Tubes/Pipes Wall d/t 50ε 2 70ε 2 90ε 2 Solid Bars Bar N/A Assumed to be Class 2 General, Section Designer Section N/A Assumed to be Class 3 Section Classification 4-7

Steel Frame Design Eurocode 3-2005 Table 4-3: Width-To-Thickness Ratios Combined Bending And Compression Shape Part Ratio Class 1 Class 2 Class 3 I-sections, Channels Web c/t 396ε /(13α 1) when α > 0.5; 36ε /α when α 0.5 Flange (tip in comp.) Flange (tip in tens.) 456ε /(13α 1) when α > 0.5; 41.5ε /α when α 0.5 c/t 9ε /α 10ε /α c/t 9ε /(α α ) 10ε /(α α ) 42ε /(0.67 + 0.33ψ) when ψ > 1; 62ε (1 ψ) ψ 1 21ε k σ ψ when Web c/t If tip is in comp. 9ε /α If tip is in tension, If tip is in comp. 10ε /α If tip is in tension, 21ε k σ Tees 9ε 10ε α α α α Flange c/t 9ε 10ε 14ε Boxes Web, flange c/t 396ε /(13α 1) when α > 0.5; 36ε /α when α 0.5 456ε /(13α 1) when α > 0.5; 41.5ε /α when α 0.5 42ε /(0.67 + 0.33ψ) when ψ > 1; 62ε (1 ψ) ψ when ψ 1 Tubes/Pipes Wall d/t 50ε 2 70ε 2 90ε 2 Solid Bars Bar N/A Assumed to be Class 2 General, Section Designer Section N/A Assumed to be Class 3 4-8 Section Classification

Chapter 5 Design for Axial Force This chapter provides a detailed description of the design algorithm for the Eurocode 3-2005 steel frame design, with respect to designing for axial forces. The following topics are covered: calculation of axial area (EC3 6.2.2) design for axial tension (EC3 6.2.3) design for axial compression (EC3 6.2.4) design for axial buckling (EC3 6.3.1) 5.1 Axial Area The gross cross-section area, A, is based on nominal dimensions, ignoring fastener holes and splice materials, and accounting for larger openings. The net cross-section area, A net, is defined as the gross cross-section area, A, minus fastener holes and other openings. By default, A net is taken equal to A. This value can be overwritten on a member-by-member basis using the Net Area to Total Area Ratio overwrite. 5-1

Steel Frame Design Eurocode 3-2005 5.2 Tension Check The axial tension check at each output station shall satisfy: N N t, Rd 1.0 (EC3 6.2.3(1)) where the design tension resistance, N t,rd is taken as the smaller of: the design plastic resistance, N pl,rd of the gross cross-section Af y N = (EC3 6.2.3(2)a) γ pl, Rd M 0 the design ultimate resistance, N u,rd of the net cross-section N = u, Rd A γ 0.9 net u M 2 f (EC3 6.2.3(2)b) The values of A and A net are defined in Section 5.1. 5.3 Compression Check The axial compression check at each output station shall satisfy: N N c, Rd 1.0 (EC3 6.2.4(1)) where the design compression resistance, N c,rd for Class 1, 2, 3, and 4 sections is taken as: Af y N = for Class 1, 2, or 3 cross-sections (EC3 6.2.4(2)) γ c, Rd c, Rd M 0 Aeff fy N = for Class 4 cross-sections (EC3 6.2.4(2)) γ M 0 5-2 Tension Check

Chapter 5 Design for Axial Force The value of A is defined in Section 5.1 of this manual. A eff is the effective area of the cross-section when subjected to uniform compression. A eff is based on the effective widths of the compression parts (EC3 6.2.9.3(2), 6.2.2.5(1)). It is determined based on the EN 1993-1-5 code (EN 1993-1-5 4.4(2), Table 4.1, Table 4.2). 5.4 Axial Buckling Check The axial buckling check at each output station shall satisfy: N N b, Rd 1.0 (EC3 6.3.1.1(1)) where the design compression resistance, N b,rd for Class 1, 2, 3, and 4 sections is taken as: N b, Rd N b, Rd χaf y = for Class 1, 2, and 3 cross-sections (EC3 6.3.1.1(3)) γ MI χaeff fy = for Class 4 cross-sections (EC3 6.3.1.1(3)) γ MI The reduction factor, χ for the relevant buckling mode is taken as: 1 χ= 1.0 2 2 Φ+ Φ λ (EC3 6.3.1.2(1)) where the factor, Φ and the non-dimensional slenderness, λ are taken as: ( ) 2 Φ= 0.5 1 +α λ 0.2 +λ (EC3 6.3.1.2(1)) Af y L λ= = N i cr cr 1, λ 1 for Class 1, 2 and 3 cross-sections (EC3 6.3.1.3(1)) A f L λ= = N i eff y cr cr A eff λ A, for Class 4 cross-sections (EC3 6.3.1.2(1)) Axial Buckling Check 5-3

Steel Frame Design Eurocode 3-2005 E λ 1 = π (EC3 6.3.1.3(1)) f y The elastic critical force, N cr is based on gross cross-section properties. The value of A is defined in Section 5.1. The imperfection factor, α is defined in Table 5.1 based on the respective buckling curve, defined in Table 5.2 of this manual. The value L cr is the effective unbraced length and i is the radius of gyration about the relevant axis. L cr is taken as follows: L cr = KL where K is the effective length factor for flexural buckling. It can assume two values: K 22 for buckling about the minor axis (z-z) and K 33 for buckling about the major axis (y-y). L is the unbraced length of the member. It also can assume two values, L 22 and L 33, for buckling about minor axis (z-z) and major axis (yy), respectively. See Sections 5.5 and 5.6 of this manual for more details on L and K. For all sections except Single Angles, the principal radii of gyration i 22 and i 33 are used. For Single Angles, the minimum (principal) radius of gyration, i,is z used instead of i 22 and i 33,conservatively, in computing Lcr i. K33 and K 22 are two values of K2 for the major and minor axes of bending. K 2 is the effective length factor for actual (sway or nonsway) conditions. The axial buckling check is not ignored if: N λ 0.2 or 0.04 (EC3 6.3.1.2(4)) N cr Table 5-1: Imperfection Factors (EC3 6.3.1.2(2), Table 6.1) Buckling Curve ao a b c d Imperfection Factor, α 0.13 0.21 0.34 0.49 0.76 5-4 Axial Buckling Check

Chapter 5 Design for Axial Force Table 5-2: Buckling Curves (EC3 6.3.1.2(2), Table 6.2) Section Shape Limits Axis Rolled I-sections Welded I-sections Hollow Tube and Pipe Sections Welded Box Channel, Tee, Double Channel, General, Solid Sections, Section Designer Angle and Double Angle Sections h/b > 1.2 h/b 1.2 tf 40 mm 40 < tf 100 mm tf 40 mm tf > 40 mm tf 100 mm tf > 100 mm Major Minor Major Minor Major Minor Major Minor Major Minor Major Minor Buckling Curve S235, S275, S355, S420 S460 hot finished any a a0 b/tf > 30 or h/tw > 30 any b b b/tf < 30 or h/tw < 30 any c c none any c c none any b b a b b c b c d d b c c d a0 a0 a a a a c c b c c d 5.5 Member Unsupported Lengths The column unsupported lengths are required to account for column slenderness effects for flexural buckling and for lateral-torsional buckling. The program automatically determines the unsupported length rations, which are specified as a fraction of the frame object length. Those ratios times the frame object lengths give the unbraced lengths for the member. Those ratios also can be overwritten by the user on a member-by-member basis, if desired, using the design overwrite option. Member Unsupported Lengths 5-5

Steel Frame Design Eurocode 3-2005 Two unsupported lengths, L 33 and L 22, as shown in Figure 5-1 are to be considered for flexural buckling. These are the lengths between support points of the member in the corresponding directions. The length L 33 corresponds to instability about the 3-3 axis (major axis), and L 22 corresponds to instability about the 2-2 axis (minor axis). The length L LTB, not shown in the figure, is also used for lateral-torsional buckling caused by major direction bending (i.e., about the 3-3 axis). Figure 5-1 Unsupported lengths L33 and L22 In determining the values for L 22 and L 33 of the members, the program recognizes various aspects of the structure that have an effect on these lengths, such as member connectivity, diaphragm constraints, and support points. The program automatically locates the member support points and evaluates the corresponding unsupported length. It is possible for the unsupported length of a frame object to be evaluated by the program as greater than the corresponding member length. For example, assume a column has a beam framing into it in one direction, but not the other, at a floor level. In that case, the column is assumed to be supported in one direction only at that story level, and its unsupported length in the other direction will exceed the story height. 5-6 Member Unsupported Lengths

Chapter 5 Design for Axial Force By default, the unsupported length for lateral-torsional buckling, L LTB, is taken to be equal to the L 22 factor. Similar to L 22 and L 33, L LTB can be overwritten. The unsupported length for minor direction bending for lateral-torsional buckling also can be defined more precisely by using precise bracing points in the Lateral Bracing option, which is accessed using the Design > Lateral Bracing command. This allows the user to define the lateral bracing of the top, bottom, or both flanges. The bracing can be a point brace or continuous bracing. The program calculates the unbraced length to determine axial capacity based on the limit state of flexural buckling from this definition. Any bracing at the top or bottom, or both, is considered enough for flexural buckling in the minor direction. While checking moment capacity for the limit state of lateraltorsional buckling (LTB) at a station, the program dynamically calculates the bracing points on the compression flange at the left and at the right of the check station considering the sign of moment diagram. This definition affects only the unbraced lengths for minor direction bending (L 22) and lateral-torsional buckling (L LTB). This exact method of bracing definition does not allow the user to define unbraced lengths for major direction bending (L 33). There are three sources of unbraced length ratio: (1) automatic calculation, (2) precise bracing definition, (3) overwrites, with increasing priority in considerations. Automatic calculation of the unbraced length is based on member connectivity considering only the members that have been entered into the model. This misses the tiny bracing members. However, such automatically calculated bracing lengths are load combo (moment diagram) independent. This can be reported easily. Similarly, the overwritten values are load combo independent. This allows the program to report the overwritten unbraced length easily. However, if the member has a precise bracing definition, the unbraced length can be different at different stations of the member along the length. Also it can be load combo dependent. Thus, when the unbraced length is reported in the detailed design info, it is reported perfectly considering all three sources as needed. However, when reporting unbraced length on the model shown in the active window, the program-reported value comes from automatic calculation or from the overwrites if the user has overwritten it. Member Unsupported Lengths 5-7

Steel Frame Design Eurocode 3-2005 5.6 Effective Length Factor (K) The effective length method for calculating member axial compressive strength has been used in various forms in several stability based design codes. The method originates from calculating effective buckling lengths, KL, and is based on elastic/inelastic stability theory. The effective buckling length is used to calculate an axial compressive strength, N b,rd, through an empirical column curve that accounts for geometric imperfections, distributed yielding, and residual stresses present in the cross-section. There are two types of K-factors in the Eurocode 3-2005 code. The first type of K-factor is used for calculating the Euler axial capacity assuming that all of the member joints are held in place, i.e., no lateral translation is allowed. The resulting axial capacity is used in calculation of the k factors. This K-factor is named as K 1 in this document. The program calculates the K 1 factor automatically based on nonsway condition. This K 1 factor is always less than 1. The program allows the user to overwrite K 1 on a member-by-member basis. The other K-factor is used for calculating the Euler axial capacity assuming that all the member joints are free to sway, i.e., lateral translation is allowed. The resulting axial capacity is used in calculating N b,rd. This K-factor is named as K 2 in this document. This K 2 is always greater than 1 if the frame is a sway frame. The program calculates the K 2 factor automatically based on sway condition. The program also allows the user to overwrite K 2 factors on a member-bymember basis. If the frame is not really a sway frame, the user should overwrite the K 2 factors. Both K 1 and K 2 have two values: one for major direction and the other for minor direction, K 1minor, K 1major, K 2minor, K 2major. There is another K-factor. K ltb for lateral-torsional buckling. By default, K ltb is taken as equal to K 2minor. However, the user can overwrite this on a member-bymember basis. Determination K 2 Factors: The K-factor algorithm has been developed for building-type structures, where the columns are vertical and the beams are horizontal, and the behavior is basically that of a moment-resisting frame for which the K-factor calculation is 5-8 Effective Length Factor (K)

Chapter 5 Design for Axial Force relatively complex. For the purpose of calculating K-factors, the objects are identified as columns, beams, and braces. All frame objects parallel to the Z- axis are classified as columns. All objects parallel to the X-Y plane are classified as beams. The remainders are considered to be braces. The beams and braces are assigned K-factors of unity. In the calculation of the K-factors for a column object, the program first makes the following four stiffness summations for each joint in the structural model: S S cx cy EI = c c Lc EI x S EI b b bx = L b x EI = c c = L S b b by c y Lb y where the x and y subscripts correspond to the global X and Y directions and the c and b subscripts refer to column and beam. The local 2-2 and 3-3 terms EI22 L 22 and EI33 L 33 are rotated to give components along the global X and Y directions to form the ( EI L ) x and ( EI L ) y values. Then for each column, the joint summations at END-I and the END-J of the member are transformed back to the column local 1-2-3 coordinate system, and the G-values for END-I and the END-J of the member are calculated about the 2-2 and 3-3 directions as follows: I S G 22 = S I S G 33 = S I c22 I b22 I c33 I b33 J S G 22 = S J G 33 = S S J c22 J b22 J c33 J b33 If a rotational release exists at a particular end (and direction) of an object, the corresponding value of G is set to 10.0. If all degrees of freedom for a particular joint are deleted, the G-values for all members connecting to that joint will be set to 1.0 for the end of the member connecting to that joint. I J Finally, if G and G are known for a particular direction, the column K 2- factor for the corresponding direction is calculated by solving the following relationship for α: Effective Length Factor (K) 5-9

Steel Frame Design Eurocode 3-2005 2 α G I 6( G G 36 = J + G ) I J α tanα (sway) from which K 2 = π/α. This relationship is the mathematical formulation for the evaluation of K-factors for moment-resisting frames assuming sidesway to be uninhibited. For other structures, such as braced frame structures, the K-factors for all members are usually unity and should be set so by the user. The following are some important aspects associated with the column K-factor algorithm: An object that has a pin at the joint under consideration will not enter the stiffness summations calculated previously. An object that has a pin at the far end from the joint under consideration will contribute only 50% of the calculated EI value. Also, beam members that have no column member at the far end from the joint under consideration, such as cantilevers, will not enter the stiffness summation. If there are no beams framing into a particular direction of a column member, the associated G-value will be infinity. If the G-value at any one end of a column for a particular direction is infinity, the K-factor corresponding to that direction is set equal to unity. If rotational releases exist at both ends of an object for a particular direction, the corresponding K-factor is set to unity. The automated K-factor calculation procedure occasionally can generate artificially high K-factors, specifically under circumstances involving skewed beams, fixed support conditions, and under other conditions where the program may have difficulty recognizing that the members are laterally supported and K-factors of unity are to be used. All K-factor produced by the program can be overwritten by the user. These values should be reviewed and any unacceptable values should be replaced. The beams and braces are assigned K-factors of unity. 5-10 Effective Length Factor (K)

Chapter 5 Design for Axial Force Determination K 1 Factors: I J If G and G are known for a particular direction, the column K 1-factor for the corresponding direction is calculated by solving the following relationship for α: ( ) ( α ) I J I J GG 2 G + G α tan α 2 α + 1 1 0 4 2 + = tan α 2 (non-sway) from which K 1 = π/α. This relationship is the mathematical formulation for the evaluation of K 1-factor for moment-resisting frames assuming sidesway to be I J inhibited. The calculation of G and G follows the same procedure as that for K 2-factor which is already described in this section. Effective Length Factor (K) 5-11

Chapter 6 Design for Bending Moment This chapter provides a detailed description of the design algorithm for the Eurocode 3-2005 steel frame design when designing for bending moments. The following topics are covered: design for bending moment (EC3 6.2.5) design for lateral-torsional buckling (EC3 6.3.2) 6.1 Moment Check The moment check at each output station shall satisfy: M M c, Rd 1.0 (EC3 6.2.5(1)) where the design moment resistance, M c,rd is taken as: Class 1 or 2 sections M Wpl fy = M = (EC3 6.2.5(2)) γ c, Rd pl, Rd M 0 6-1

Steel Frame Design Eurocode 3-2005 Class 3 sections M W = M = γ c, Rd el. Rd el,min M 0 f y (EC3 6.2.5(2)) Class 4 sections: W M c, Rd = γ eff,min M 0 f y (EC3 6.2.5(2)) The plastic and elastic section modulus values, W pl and W el,min are part of the frame section definition. W eff,min is the effective section modulus, corresponding to the fiber with the maximum elastic stress, of the cross-section when subjected only to moment about the relevant axis. W eff,min is based on the effective widths of the compression parts (EC3 6.2.9.3(2), 6.2.2.5(1)). It is determined based on EN 1993-1-5 code (EN 1993-1-5 4.4(2), Table 4.1, Table 4.2). The effect of high shear on the design moment resistance, M c,rd is considered if: V 0.5V (EC3 6.2.8(2)) pl, Rd To account for the effect of high shear in I-sections, Channels, Double Channels, Rectangular Hollow Sections, Tee and Double Angle sections subjected to major axis moment, the reduced design plastic resistance moment is taken as: M nρa 2 w Wpl, y fy 4tw y, V, Rd = M y, c, Rd γm 0 (EC3 6.2.8(5)) where n, ρ and A w are taken as: 1 for I, Channel, and Tee sections n = 2 for Double Channel, Hollow Rectangular, and (EC3 6.2.8(5)) Double Angle sections 6-2 Moment Check

Chapter 6 Design for Bending Moment 2V ρ= 1 V pl, Rd 2 (EC3 6.2.8(3)) A w = ht (EC3 6.2.8(5)) w w For all other sections, including Hollow Pipe, Solid Rectangular, Circular, and Angle sections, the reduced design plastic resistance moment is taken as: M y,v,rd = (1 ρ) M c,rd (EC3 6.2.8(3)) Similarly, for I, Channel, Double Channel, Rectangular Hollow, Tee and Double Angle sections, if the minor direction shear is more than 0.5 times the plastic shear resistance in the minor direction, the corresponding plastic resistance moment is also reduced as follows: M nρa 2 f Wpl, z fy 4t f z, V, Rd = MZ, c, Rd γm 0 (EC3 6.2.8(5)) where, 1, for Tee sections, 2, for I, Channel, Double Angle, n = (EC3 6.2.8(5)) Hollow Rectangular sections, and 4, for Double Channel sections. 2V y, ρ= 1 Vy, pl, Rd 2 (EC3 6.2.8(3)) A = bt (EC3 6.2.8(5)) f f f For all other sections, the reduced design plastic resistance moment is taken as: M ( 1 ) = ρ M (EC3 6.2.8(3)) z, V, Rd z, c, Rd Moment Check 6-3

Steel Frame Design Eurocode 3-2005 6.2 Lateral-Torsional Buckling Check The lateral-torsional buckling check at each output station shall satisfy: M M b, Rd 1.0 (EC3 6.3.2.1(1)) where the design buckling resistance moment, M b,rd is taken as: M = χ W b, Rd LT y λ f y MI (EC3 6.3.2.1(3)) and the section modulus, W y is defined based on the section classification: Class 1 or 2 sections W y = W (EC3 6.3.2.1(3)) pl, y Class 3 sections W y = W (EC3 6.3.2.1(3)) el, y Class 4 sections W W (EC3 6.3.2.1(3)) y = eff, y W pl, W el, and W eff have been described in the previous section. The reduction factor χ LT is taken as: 1 χ LT = 1.0 Φ + Φ λ 2 2 LT LT LT (EC3 6.3.2.2(1)) where the factor, Φ, and the non-dimensional slenderness, λ LT are taken as: ( ) 2 Φ LT = 0.5 1 +αlt λlt 0.2 +λ LT (EC3 6.3.2.2(1)) Wy fy λ LT = (EC3 6.3.2.2(1)) M 6-4 Lateral-Torsional Buckling Check cr